More problems from Liboff chapter 4. Hermitian operators.
Posted by peeterjoot on June 27, 2010
Some more problems from .
Some problems on Hermitian adjoints. The starting point is the definition of the adjoint of in terms of the inner product
Hermitian adjoint of , where . Here we need the integral form of the inner product
Since the text shows that the square of a Hermitian operator is Hermitian, one perhaps wonders if is (but we expect not since is Hermitian).
Suppose , we have
so for this to be Hermitian () we must have . If , we have
So , and . This fixes the scalar multiples of that are required to form a Hermitian operator
where is any real positive constant.
This one was to calculate . Intuitively I’d expect that . How could one show this?
Trying to show this with Dirac notation, I got all mixed up initially.
Using the more straightforward and old fashioned integral notation (as in ), this is more straightforward. We have the Hermitian conjugate defined by
Or, more symmetrically, using braces to indicate operator direction
Introduce a couple of variable substuitions for clarity
We then have
Since this is true for all , we have as expected.
Having figured out the problem in the simpleton way, it’s now simple to go back and translate this into the Dirac inner product notation without getting muddled. We have
Problem 4.12 d
If is not Hermitian, is the product Hermitian? To start we need to verify that .
With that verified we have
so, the answer is yes. Provided the adjoint exists, that product will be Hermitian.
Show that (that it is real), if is Hermitian. This follows by expansion of that conjuagate
 R. Liboff. Introductory quantum mechanics. 2003.
 D. Bohm. Quantum Theory. Courier Dover Publications, 1989.